Wavefront Solutions for Reaction-diffusion-convection Models with Accumulation Term and Aggregative Movements
Marco Cantarini, Cristina Marcelli, Francesca Papalini
TL;DR
The paper studies traveling wave propagation in a reaction-diffusion-convection system with a degenerate accumulation term and diffusivity that can change sign. It reduces the traveling wave problem to a first-order singular boundary-value problem for z(u)=D(u)u'(t), proves a threshold speed hat c above which traveling waves exist and below which they do not, and classifies waves into classical and sharp types. The authors handle sign-changing diffusivity by decomposing (0,1) into sign-consistent intervals and applying upper/lower-solution techniques to glue local solutions, providing explicit bounds for hat c and illustrating with examples. The results extend prior work with positive diffusivity and illuminate wave selection and finite-speed propagation in heterogeneous media with aggregation movements.
Abstract
In this paper we analyze the wavefront solutions of parabolic partial differential equations of the type \[ g(u)u_τ+f(u)u_{x}=\left(D(u)u_{x}\right)_{x}+ρ(u),\quad u\left(τ,x\right)\in[0,1] \] where the reaction term $ρ$ is of monostable-type. We allow the diffusivity $D$ and the accumulation term $g$ to have a finite number of changes of sign. We provide an existence result of travelling wave solutions (t.w.s.) together with an estimate of the threshold wave speed. Finally, we classify the t.w.s. between classical and sharp ones.